Summary
This paper looks at how to better predict whether an infection will keep spreading or die out, based on early observations. The authors focus on a model that considers differences across a population and movement of infected individuals. They find that the key factors controlling persistence are the difference between how fast the disease spreads and how fast people recover, along with how infected individuals move around. Using Bayesian statistics, they show it’s possible to estimate the important disease threshold even with uncertain data, and they prove their methods work mathematically. They also provide examples showing how close their predictions are when infections start at low levels.
What this means in practice
- •For epidemic modelers: Estimate the likelihood of disease persistence from early, noisy data in spatially heterogeneous populations using a Bayesian framework.
- •For public health analysts: Provide better estimates of epidemic thresholds considering spatial movement and population differences to inform intervention timing.
Abstract
We study Bayesian inference of persistence from early observations of a heterogeneous susceptible--infected--susceptible (SIS) reaction--diffusion model with standard incidence. We show that the leading prevalence dynamics depend on the unknown coefficients only through the difference between the transmission and recovery rates and the infected diffusivity. The same pair determines the principal eigenvalue whose sign governs persistence, so early prevalence can inform the threshold even when the individual rates remain uncertain. We prove global weak well-posedness and quantitative parameter stability for the SIS system and use the resulting observation-map estimates to establish Bayesian well-posedness. For an initial infection of size $\varepsilon$, we derive a uniform second-order expansion that describes how the separate rates enter prevalence. Incidence depends explicitly on transmission at leading order. When the noise standard deviation is proportional to $\varepsilon$, we prove $O(\varepsilon)$ convergence in Hellinger distance of the nonlinear posteriors to their low-prevalence limits for both prevalence alone and joint prevalence and incidence data. We bound the principal eigenvalue using the growth rate of the infected $L^2$ norm between two times. This rate gives a lower bound for the full nonlinear system, and the upper bound separates spatial transients from nonlinear saturation. Uniform spectral estimates and bounds on infection size and measurement error give sufficient conditions for persistence and extinction. We also bound the error in posterior persistence probabilities under forward and eigenvalue approximation, accounting for posterior mass near the threshold. Numerical examples illustrate threshold inference and the predicted \(O(\varepsilon)\) convergence of the nonlinear posteriors to their low-prevalence limits in Hellinger distance.